The weight two \(K\)-theory of fields
type: Item
instance of: scholarly article
title: The weight two \(K\)-theory of fields
zbMATH Open document ID: 0837.19004
DOI: 10.1007/BF00961280
published in: \(K\)-Theory
publication date: 1996-05-14T00:00:00Z
review text: Let \(K_p(F, \mathbb{Z}/n)\) denote the \(K\)-groups of a field \(F\), with finite coefficients. For \((n,\text{char } F)= 1\), there are Chern class maps \[ c_{q, p}: \text{gr}^q_\gamma K_{2q- p}(F, \mathbb{Z}/n)\to H^p_{\text{ét}}(F, \mu^{\otimes q}_n); \] it is expected (a version of the Quillen-Lichtenbaum conjectures) that the maps \(c_{q,p}\) should be isomorphisms after inverting ``small'' primes. The author proves (theorem 6.8) that \(c_{2,p}\) are isomorphisms after inverting \((3- p)!\), \(p= 2, 1, 0, -1,\dots\) and for \((n, \text{char } F)= 1\). It implies, when \(F\) is a number field, that \[ \text{gr}^2_\gamma K_p(F)[1/(p- 1)!]= 0\qquad\text{for}\quad p\geq 4 \] and suggests the refined version of the conjecture of Soulé and Beilinson: \[ \text{gr}^q_\gamma K_p(F)[1/(p- 1)!]= 0\qquad\text{for}\quad p\geq 2q. \] Theorem 6.8 is deduced from Hilbert's Theorem 90 for the relative \(K_2\) of regular semi-local rings containing a field, the main technical result of the paper.
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 12F99
zbMATH DE Number: 827887
zbMATH Keywords: étale cohomology
zbMATH Keywords: gamma filtration
zbMATH Keywords: \(K_ 3\)
zbMATH Keywords: \(K_ 2\)
zbMATH Keywords: Chern class maps
zbMATH Keywords: Quillen-Lichtenbaum conjectures
author: Marc Levine
MaRDI profile type: Publication
cites work: Algebraic cycles and higher K-theory
cites work: Quillen's K-theory and algebraic cycles on almost non-singular varieties
cites work: Multirelative algebraic K-theory: The group \(K_ 2(\Lambda;I_ 1,\dots ,I_ n)\) and related computations
cites work: Continuous étale cohomology
cites work: The $K_2$ of rings with many units
cites work: Q3475449
cites work: The relativization of \(K_2\)
cites work: The indecomposable $K_3$ of fields
cites work: Relative Milnor \(K\)-theory
cites work: Values of zeta-functions at non-negative integers
cites work: Cohomologie et groupe de Steinberg rélatifs
cites work: $ K$-COHOMOLOGY OF SEVERI-BRAUER VARIETIES AND THE NORM RESIDUE HOMOMORPHISM
cites work: THE GROUP $ K_3$ FOR A FIELD
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: On the cohomology and K-theory of the general linear groups over a finite field
cites work: K-cohomology of regular schemes
cites work: Opérations En K-Théorie Algébrique
cites work: K-théorie des anneaux d'entiers de corps de nombres et cohomologie etale
cites work: Torsion in \(K_ 2\) of fields
cites work: Relations between \(K_2\) and Galois cohomology
cites work: KV-Theory of Categories
description: scientific article; zbMATH DE number 827887